Closed Loop Frequency Response The Bode plot is generally constructed for an open loop transfer function of a system. In order to draw the Bode plot for a closed loop system, the transfer function has to be developed, and then decomposed into its poles and zeros. This process is tedious and cannot be carried out without the ais of a very powerful calculator or a computer. Nichols, developed a simple process by which the unity feedback closed loop frequency response of a system can be easily deduced from the open loop transfer function. His approach will be outlined on the next few slides. While the Nichols approach is derived strictly for a unity feedback system, any non-unity feedback system can be transformed to two cascaded open loop systems as follows:
Closed Loop Frequency Response Consider the following non unity feedback system: R(s) + - G(s) C(s) H(s) It can be transformed into the following system composed of a simple block cascaded to a unity feedback system: R(s) 1/H(s) + - G(s) C(s) The frequency response can be then obtained using the additive feature of the Bode plots.
Closed Loop Frequency Response With reference to the generic unity feedback system block diagram and its polar plot; the system transfer function is given by: R(s) + - G(s) C(s) ( ) ( ) C s R s ( ) ( ) G s = 1 + G s The dashed line in the polar plot is the trace of the tip of the vector OA which represents the system. The length of the vector measure the magnitude of the system at a given frequency T, and the angle N represents the phase shift. P (-1,0) M "=N- A Polar Plot Im O N Re
Closed Loop Frequency Response With reference to the polar plot; OA represents G j OP represents -1 But PA= PO+ OA = 1+ OA G = PA 1 + G ( ω ) G( jω ) ( jω ) ( jω ) P (-1,0) M "=N- A Polar Plot Im O N Re The phase shift angle of the closed loop system is the included angle between the vectors OA and PA, i.e.; α = φ θ
Closed Loop Constant Magnitude Locii Since the open loop transfer function is a complex quantity that can be expressed as: G( jω ) = X + j It follows that the magnitude of the closed loop system can be expressed as follows: M = = G 1 + ( jω ) ( jω ) G X + ( 1 X ) j + + j M = X + ( 1 X ) + +
Closed Loop Constant Magnitude Locii Expanding and collecting terms, the previous equation gives: ( ) ( ) X 1 M M X M + 1 M = 0 For M=1 the above equation reduces to; X = 1 For M 1, the equation can be written in the form: M M X + X M 1 + M 1 + = 0
Closed Loop Constant Magnitude Locii The preceding equation canot be factored as it stands. However by adding; M ( M 1) to both sides and factoring gives: M M X+ + = M 1 1 ( M ) This is an equation of a circle with radius and center as follows: center: M M, 0 and radius: M 1 M 1
Closed Loop Constant Magnitude Locii The plot of the previous equation is shown below for different magnitudes M:
Closed Loop Constant Phase Locii The phase of the closed loop system can be expressed as follows: jα X+ j e = 1 + X + j α = X 1 1 tan tan 1 + X Now let " = N, then; N = X 1 1 tan tan tan 1 + X
Closed Loop Constant Phase Locii But tan tan A tan B = 1 + tan A tanb ( A B) ( ) ( ) X 1+ X ( )( ) N = 1 + X 1+ X Simplifying gives: N = X + X + or 1 + + = N X X 0
Closed Loop Constant Phase Locii The preceding equation canot be factored as it stands. However by adding; 1 + 1 4 N ( ) to both sides and factoring gives: 1 1 1 1 X+ + = + N 4 N This is an equation of a circle with radius and center as follows: 1 1 1 1 center:, and radius: + N 4 ( N )
Closed Loop Constant Phase Locii The plot of the previous equation is shown below for different phase angles ":
Closed Loop Frequency Response from Open Loop Response Using the constant magnitude and phase circles and the polar plot of the open loop to obtain the closed loop frequency response. ( ) G s = 50 ( + 3)( s+ 6) s s
The Nichols Chart Nichols superimposed the log of the constant magnitude and phase circles on the logmagnitude phase plot to create the Nichols chart. The magnitude and phase of the open loop values are plotted on the rectilinear logmagnitude phase grid, and the closed loop frequency response is read from the curvilinear grid.
The Nichols Chart ( ) Gjω = 1 jω jω j ( + 1)( 0. ω+ 1)
The Nichols Chart ( ω) G j = 0.64 ( 1+ j ) jω ω ω